In this article, we study and settle several structural questions concerning the exact solvability of the Olshanetsky-Perelomov quantum Hamiltonians corresponding to an arbitrary root system. We show that these operators can be written as linear combinations of certain basic operators admitting infinite flags of invariant subspaces, namely the Laplacian and the logarithmic gradient of invariant factors of the Weal denominator. The coefficients of the constituent linear combination become the coupling constants of the final model. We also demonstrate the L~2 completeness of the eigenfunctions obtained by this procedure, and describe a straightforward recursive procedure based on the Freudenthal multiplicity formula for constructing the eigenfunctions explicitly.
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